Regularity index of fat points in the projective plane
type: Item
instance of: scholarly article
title: Regularity index of fat points in the projective plane
zbMATH Open document ID: 0821.14031
author: Giuliana Fatabbi
published in: Journal of Algebra
publication date: 1995-10-03T00:00:00Z
review text: The aim of this paper is to find a bound for the minimum value of \(t > 0\) for which a given scheme of fat points in \(\mathbb{P}^ 2\) imposes independent conditions on curves of degree \(t\). A scheme \(X\) of ``fat points'' is a 0-dimensional scheme defined by a homogeneous ideal \(I = {\mathfrak p}_ 1^{m_ 1} \cap {\mathfrak p}_ 2^{m_ 2} \cap \cdots \cap {\mathfrak p}_ s^{m_ s} \subseteq k [x_ 0,x_ 1,x_ 2]\), where \(k\) is an algebraically closed field with \(\text{char} k = 0\). Every vector space \(I_ t\) represents the curves of degree \(t\) having multiplicity at least \(m_ i\) at each \(P_ i\), \(i = 1,\dots, s\). Let \(\tau = \tau (I)\) be the index of regularity of \(I\) (or of \(X)\), i.e. the minimum value of \(t\) for which \(X\) imposes independent conditions on curves of degree \(t\) (equivalently, the minimum \(t\) for which \(H^ 1 (\mathbb{P}^ 2, {\mathcal I} (t)) = 0\), where \({\mathcal I}\) is the ideal sheaf of \(X\), or also the minimum \(t\) for which \(H(X,t) = \deg X = \sum {m_ i + 1 \choose 2}\), where \(H(X,.)\) is the Hilbert function of \(X)\). -- The most general upper bound possible for \(\tau\) is: \(\tau \leq \sum m_ i - 1\), which is attained when the \(s\) points are collinear. If we suppose that the points are in general position (no three on a line) and that \(m_ 1 \leq \cdots \leq m_ s\), then a better bound has been given by \textit{B. Segre}: \(\tau \leq \max \{m_ 1 + m_ 2 - 1\), \([\sum m_ i/2]\}\). In this paper a bound is given that generalizes the two mentioned above; for any scheme \(X\) of fat points, we have \(\tau \leq \max \{h - 1, [\sum m_ i/2]\}\), where \(h\) is the maximum among the sums of multiplicities taken over all subsets of collinear points. Moreover, if \(\max\{h - 1, [\sum m_ i/2]\} = h - 1\), then \(\tau = h - 1\).
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 705010
zbMATH Keywords: points in general position
zbMATH Keywords: fat points
zbMATH Keywords: index of regularity
zbMATH Keywords: Hilbert function
MaRDI profile type: Publication
full work available at URL: jabr.1994.1370
OpenAlex ID: W2032113726
DOI: 10.1006/JABR.1994.1370
description: scientific article; zbMATH DE number 705010